Topology and its Applications

Content

During July 10-28, 2006 , Mississippi State University, Starkville will be the host of the Institute for Mathematics and its Applications (IMA) summer graduate program in mathematics. The course will concentrate on Topology and its Applications.

This program is open to graduate students from IMA Participating Institutions. Students are nominated by their department head. Participating institution department heads nominate graduate students from their institution by an e-mail to visit@ima.umn.edu with the students' names and e-mail addresses.

Those students may then register by filling out the application form. Places are guaranteed for two graduate students from each participating institution, with additional students accommodated as space allows.

Course Description:

In a number of diverse areas, topological issues have begun to surface. In molecular biology, for example, the geometric features of the surface of a molecule have been shown to influence certain protein docking processes. Knot theory is becoming increasingly important in the study of DNA. Computer scientists encounter topological problems in attempts to reconstruct surfaces from sampled data. Topology in phase space can help overcome the inherent sensitivity in longtime simulations of dynamical systems. During the three-week meeting in the period July 9-29, 2006, there will be three week-long courses in the following areas:

We will discuss computational homology and its use in the study of nonlinear dynamical systems. The lectures will survey five topics:(1) Algorithms for computing the homology of spaces.(2) The use of homology in investigating and classifying nonlinear systems based on the patterns observed in experiments or numerical simulations.(3) Algorithms for computing the homology of maps.(4) Conley index theory and associated algorithms.(5) Computer assisted proofs in dynamics.The associated projects will involve the application of these tools to specific problems.

High dimensional data is now being generated at a very rapid rate in many different disciplines. Further, the data is frequently noisy, and is not equipped with any theoretical model. Rather, the data analysis needs to be used to discover the model. Since there is frequently no model, and therefore no preferred coordinate system, it is important to study those properties which don't change under continuous coordinate changes, which are called topological. The goal of this course is to provide an introduction to a recently developed computational version of algebraic topology, called persistent homology, which allows one to infer topological properties of geometric objects from "point clouds" sampled from them. We will introduce algebraic topology itself, the theory of persistent homology, software which permits its computation, and demonstrate how it is used in several real world examples.

Applications to Molecular Biology (Week 3): John Harer, Duke University